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<title>qdelaunay Qu -- furthest-site Delaunay triangulation</title>
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<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><img
src="qh--dt.gif" alt="[delaunay]" align="middle" width="100"
height="100"></a>qdelaunay Qu -- furthest-site Delaunay triangulation</h1>

<p>The furthest-site Delaunay triangulation corresponds to the upper facets of the <a href="qdelaun.htm">Delaunay construction</a>.
Its vertices are the
extreme points of the input sites.
It is the dual of the <a
href="qvoron_f.htm">furthest-site Voronoi diagram</a>.

<blockquote>
<dl>
    <dt><b>Example:</b> rbox 10 D2 | qdelaunay <a
        href="qh-optq.htm#Qu">Qu</a> <a
        href="qh-optq.htm#Qt">Qt</a> <a href="qh-opto.htm#s">s</a>
        <a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
        result</a></dt>
    <dd>Compute the 2-d, furthest-site Delaunay triangulation of 10 random
        points. Triangulate the output.
        Write a summary to the console and the regions to
        'result'.</dd>
    <dt>&nbsp;</dt>
    <dt><b>Example:</b> rbox 10 D2 | qdelaunay <a
        href="qh-optq.htm#Qu">Qu</a> <a
        href="qh-optq.htm#QJ">QJ</a> <a href="qh-opto.htm#s">s</a>
        <a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
        result</a></dt>
    <dd>Compute the 2-d, furthest-site Delaunay triangulation of 10 random
        points. Joggle the input to guarantee triangular output.
        Write a summary to the console and the regions to
        'result'.</dd>
    <dt>&nbsp;</dt>
    <dt><b>Example:</b> rbox r y c G1 D2 | qdelaunay <a
        href="qh-optq.htm#Qu">Qu</a> <a href="qh-opto.htm#s">s</a>
        <a href="qh-optf.htm#Fv">Fv</a> <a href="qh-optt.htm#TO">TO
        result</a></dt>
    <dd>Compute the 2-d, furthest-site Delaunay triangulation of a triangle inside
                a square.
        Write a summary to the console and unoriented regions to 'result'.
        Merge regions for cocircular input sites (e.g., the square).
                The square is the only furthest-site
        Delaunay region.</dd>
</dl>
</blockquote>

<p>As with the Delaunay triangulation, Qhull computes the
furthest-site Delaunay triangulation by lifting the input sites to a
paraboloid. The lower facets correspond to the Delaunay
triangulation while the upper facets correspond to the
furthest-site triangulation. Neither triangulation includes
&quot;vertical&quot; facets (i.e., facets whose last hyperplane
coefficient is nearly zero). Vertical facets correspond to input
sites that are coplanar to the convex hull of the input. An
example is points on the boundary of a lattice.</p>

<p>By default, qdelaunay merges cocircular and cospherical regions.
For example, the furthest-site Delaunay triangulation of a square inside a diamond
('rbox D2 c d G4 | qdelaunay Qu') consists of one region (the diamond).

<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output),
all furthest-site Delaunay regions will be simplicial (e.g., triangles in 2-d).
Some regions may be
degenerate and have zero area.

<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input), all furthest-site
Delaunay regions
will be simplicial (e.g., triangles in 2-d).  Joggled input
is less accurate than triangulated output ('Qt').  See <a
href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>

<p>The output for 3-d, furthest-site Delaunay triangulations may be confusing if the
input contains cospherical data. See the FAQ item
<a href=qh-faq.htm#extra>Why
are there extra points in a 4-d or higher convex hull?</a>
Avoid these problems with triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') or
joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
</p>

<p>The 'qdelaunay' program is equivalent to
'<a href="qhull.htm#outputs">qhull d</a> <a href="qh-optq.htm#Qbb">Qbb</a>'.  It disables the following Qhull
<a href=qh-quick.htm#options>options</a>: <i>d n v H U Qb QB Qc Qf Qg Qi
Qm Qr Qv Qx TR E V FC Fi Fo Fp Ft FV Q0,etc</i>.


<p><b>Copyright &copy; 1995-2020 C.B. Barber</b></p>

<hr>

<h3><a href="#TOP">&#187;</a><a name="synopsis">furthest-site qdelaunay synopsis</a></h3>
<blockquote>

See <a href="qdelaun.htm#synopsis">qdelaunay synopsis</a>.  The same
program is used for both constructions.  Use option '<a href="qh-optq.htm#Qu">Qu</a>'
for furthest-site Delaunay triangulations.

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="input">furthest-site qdelaunay
input</a></h3>
<blockquote>

<p>The input data on <tt>stdin</tt> consists of:</p>
<ul>
    <li>dimension
    <li>number of points</li>
    <li>point coordinates</li>
</ul>

<p>Use I/O redirection (e.g., qdelaunay Qu &lt; data.txt), a pipe (e.g., rbox 10 | qdelaunay Qu),
or the '<a href="qh-optt.htm#TI">TI</a>' option (e.g., qdelaunay Qu TI data.txt).

<p>For example, this is a square containing four random points.
Its furthest-site Delaunay
triangulation contains one square.
<p>
<blockquote>
<tt>rbox c 4 D2 &gt; data</tt>
<blockquote><pre>
2 RBOX c 4 D2
8
-0.4999921736307369 -0.3684622117955817
0.2556053225468894 -0.0413498678629751
0.0327672376602583 -0.2810408135699488
-0.452955383763607 0.17886471718444
  -0.5   -0.5
  -0.5    0.5
   0.5   -0.5
   0.5    0.5
</pre></blockquote>

<p><tt>qdelaunay Qu i &lt; data</tt>
<blockquote><pre>

Furthest-site Delaunay triangulation by the convex hull of 8 points in 3-d:

  Number of input sites: 8
  Number of Delaunay regions: 1
  Number of non-simplicial Delaunay regions: 1

Statistics for: RBOX c 4 D2 | QDELAUNAY s Qu i

  Number of points processed: 8
  Number of hyperplanes created: 20
  Number of facets in hull: 11
  Number of distance tests for qhull: 34
  Number of merged facets: 1
  Number of distance tests for merging: 107
  CPU seconds to compute hull (after input): 0.02

1
7 6 4 5
</pre></blockquote>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="outputs">furthest-site qdelaunay
outputs</a></h3>
<blockquote>

<p>These options control the output of furthest-site Delaunay triangulations:</p>
<blockquote>

<dl compact>
    <dd><b>furthest-site Delaunay regions</b></dd>
    <dt><a href="qh-opto.htm#i">i</a></dt>
    <dd>list input sites for each furthest-site Delaunay region. The first line is the number of regions.  The
        remaining lines list the input sites for each region.  The regions are
                oriented.  In 3-d and
        higher, report cospherical sites by adding extra points.  For the points-in-square example,
        the square is the only furthest-site Delaunay region.</dd>
    <dt><a href="qh-optf.htm#Fv">Fv</a></dt>
    <dd>list input sites for each furthest-site Delaunay region.  The first line is the number of regions.
        Each remaining line starts with the number of input sites.  The regions
        are unoriented.  For the points-in-square example,
        the square is the only furthest-site Delaunay region.</dd>
    <dt><a href="qh-optf.htm#Ft">Ft</a></dt>
    <dd>print a triangulation of the furthest-site Delaunay regions in OFF format.  The first line
        is the dimension.  The second line is the number of input sites and added points,
        followed by the number of simplices and the number of ridges.
    The input coordinates are next, followed by the centrum coordinates.  There is
        one centrum for each non-simplicial furthest-site Delaunay region.  Each remaining line starts
        with dimension+1.  The
        simplices are oriented.
        For the points-in-square example, the square has a centrum at the
        origin.  It splits the square into four triangular regions.</dd>
    <dt><a href="qh-optf.htm#Fn">Fn</a></dt>
    <dd>list neighboring regions for each furthest-site Delaunay region.  The first line is the
        number of regions.  Each remaining line starts with the number of
        neighboring regions.  Negative indices (e.g., <em>-1</em>) indicate regions
        outside of the furthest-site Delaunay triangulation.
        For the points-in-square example, the four neighboring regions
        are outside of the triangulation.  They belong to the regular
        Delaunay triangulation.</dd>
    <dt><a href="qh-optf.htm#FN">FN</a></dt>
    <dd>list the furthest-site Delaunay regions for each input site.  The first line is the
        total number of input sites.  Each remaining line starts with the number of
        furthest-site Delaunay regions.  Negative indices (e.g., <em>-1</em>) indicate regions
        outside of the furthest-site Delaunay triangulation.
        For the points-in-square example, the four random points belong to no region
        while the square's vertices belong to region <em>0</em> and three
        regions outside of the furthest-site Delaunay triangulation.</dd>
    <dt><a href="qh-optf.htm#Fa">Fa</a></dt>
    <dd>print area for each furthest-site Delaunay region. The first line is the number of regions.
        The areas follow, one line per region.  For the points-in-square example, the
        square has unit area. </dd>

    <dt>&nbsp;</dt>
    <dt>&nbsp;</dt>
    <dd><b>Input sites</b></dd>
    <dt><a href="qh-optf.htm#Fx">Fx</a></dt>
    <dd>list extreme points of the input sites.  These points are vertices of the furthest-point
        Delaunay triangulation.  They are on the
        boundary of the convex hull.   The first line is the number of
        extreme points.  Each point is listed, one per line.  The points-in-square example
        has four extreme points.</dd>
    <dt>&nbsp;</dt>
    <dt>&nbsp;</dt>
    <dd><b>General</b></dd>
    <dt><a href="qh-optf.htm#FA">FA</a></dt>
    <dd>compute total area for '<a href="qh-opto.htm#s">s</a>'
        and '<a href="qh-optf.htm#FS">FS</a>'.  This is the
                same as the area of the convex hull.</dd>
    <dt><a href="qh-opto.htm#o">o</a></dt>
    <dd>print upper facets of the corresponding convex hull (a
        paraboloid)</dd>
    <dt><a href="qh-opto.htm#m">m</a></dt>
    <dd>Mathematica output for the upper facets of the paraboloid (2-d triangulations).</dd>
    <dt><a href="qh-optf.htm#FM">FM</a></dt>
    <dd>Maple output for the upper facets of the paraboloid (2-d triangulations).</dd>
    <dt><a href="qh-optg.htm#G">G</a></dt>
    <dd>Geomview output for the paraboloid (2-d or 3-d triangulations).</dd>
    <dt><a href="qh-opto.htm#s">s</a></dt>
    <dd>print summary for the furthest-site Delaunay triangulation. Use '<a
        href="qh-optf.htm#Fs">Fs</a>' and '<a
        href="qh-optf.htm#FS">FS</a>' for numeric data.</dd>
</dl>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="controls">furthest-site qdelaunay
controls</a></h3>
<blockquote>

<p>These options provide additional control:</p>
<blockquote>

<dl compact>
   <dt><a href="qh-optq.htm#Qu">Qu</a></dt>
    <dd>must be used for furthest-site Delaunay triangulation.</dd>
    <dt><a href="qh-optq.htm#Qt">Qt</a></dt>
    <dd>triangulated output.  Qhull triangulates non-simplicial facets.  It may produce
     degenerate facets of zero area.</dd>
   <dt><a href="qh-optq.htm#QJn">QJ</a></dt>
    <dd>joggle the input to avoid cospherical and coincident
        sites.    It is less accurate than triangulated output ('Qt').</dd>
    <dt><a href="qh-optq.htm#QRn">QRn</a></dt>
    <dd>randomly rotate the input with a random seed of n.  If n=0, the
        seed is the time.  If n=-1, use time for the random seed, but do
        not rotate the input.</dd>
    <dt><a href="qh-optq.htm#QVn">QVn</a></dt>
    <dd>select facets adjacent to input site <em>n</em> (marked
        'good').</dd>
    <dt><a href="qh-optt.htm#Tv">Tv</a></dt>
    <dd>verify result.</dd>
    <dt><a href="qh-optt.htm#TI">TI file</a></dt>
    <dd>input data from file.  The filename may not use spaces or quotes.</dd>
    <dt><a href="qh-optt.htm#TO">TO file</a></dt>
    <dd>output results to file.  Use single quotes if the filename
        contains spaces (e.g., <tt>TO 'file with spaces.txt'</tt></dd>
    <dt><a href="qh-optt.htm#TFn">TFn</a></dt>
    <dd>report progress after constructing <em>n</em> facets</dd>
    <dt><a href="qh-optp.htm#PDk">PDk:1</a></dt>
    <dd>include upper and lower facets in the output.  Set <em>k</em>
        to the last dimension (e.g., 'PD2:1' for 2-d inputs). </dd>
    <dt><a href="qh-opto.htm#f">f</a></dt>
    <dd>facet dump.  Print the data structure for each facet (i.e., furthest-site Delaunay region).</dd>
</dl>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="graphics">furthest-site qdelaunay
graphics</a></h3>
<blockquote>

See <a href="qdelaun.htm#graphics">Delaunay graphics</a>.
They are the same except for Mathematica and Maple output.

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="notes">furthest-site
qdelaunay notes</a></h3>
<blockquote>

<p>The furthest-site Delaunay triangulation does not
record coincident input sites.  Use <tt>qdelaunay</tt> instead.

<p><tt>qdelaunay Qu</tt> does not work for purely cocircular
or cospherical points (e.g., rbox c | qdelaunay Qu).  Instead,
use <tt>qdelaunay Qz</tt> -- when all points are vertices of the convex
hull of the input sites, the Delaunay triangulation is the same
as the furthest-site Delaunay triangulation.

<p>A non-simplicial, furthest-site Delaunay region indicates nearly cocircular or
cospherical input sites. To avoid non-simplicial regions triangulate
the output ('<a href="qh-optq.htm#Qt">Qt</a>') or joggle
the input ('<a href="qh-optq.htm#QJn">QJ</a>').  Joggled input
is less accurate than triangulated output.
You may also triangulate
non-simplicial regions with option '<a
href="qh-optf.htm#Ft">Ft</a>'. It adds
the centrum to non-simplicial regions. Alternatively, use an <a
href="qh-impre.htm#exact">exact arithmetic code</a>.</p>

<p>Furthest-site Delaunay triangulations do not include facets that are
coplanar with the convex hull of the input sites.  A facet is
coplanar if the last coefficient of its normal is
nearly zero (see <a href="http://www.qhull.org/src/libqhull_r/user_r.h#ZEROdelaunay">qh_ZEROdelaunay</a>).

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="conventions">furthest-site qdelaunay conventions</a></h3>
<blockquote>

<p>The following terminology is used for furthest-site Delaunay
triangulations in Qhull. The underlying structure is the upper
facets of a convex hull in one higher dimension. See <a
href="qconvex.htm#conventions">convex hull conventions</a>, <a
href="qdelaun.htm#conventions">Delaunay conventions</a>,
and <a href="index.htm#structure">Qhull's data structures</a></p>
<blockquote>
<ul>
    <li><em>input site</em> - a point in the input (one dimension
        lower than a point on the convex hull)</li>
    <li><em>point</em> - <i>d+1</i> coordinates. The last
        coordinate is the sum of the squares of the input site's
        coordinates</li>
    <li><em>vertex</em> - a point on the paraboloid. It
        corresponds to a unique input site. </li>
    <li><em>furthest-site Delaunay facet</em> - an upper facet of the
        paraboloid. The last coefficient of its normal is
                clearly positive.</li>
    <li><em>furthest-site Delaunay region</em> - a furthest-site Delaunay
            facet projected to the input sites</li>
    <li><em>non-simplicial facet</em> - more than <em>d</em>
        points are cocircular or cospherical</li>
    <li><em>good facet</em> - a furthest-site Delaunay facet with optional
        restrictions by '<a href="qh-optq.htm#QVn">QVn</a>', etc.</li>
</ul>
</blockquote>
</blockquote>
<h3><a href="#TOP">&#187;</a><a name="options">furthest-site qdelaunay options</a></h3>
<blockquote>

See <a href="qdelaun.htm#options">qdelaunay options</a>.    The same
program is used for both constructions.  Use option '<a href="qh-optq.htm#Qu">Qu</a>'
for furthest-site Delaunay triangulations.

</blockquote>
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